0
votes
1answer
28 views

A notational issue with manifold coordinate charts.

Untidy notations in differential geometry is killing me. There are many instances of this but I really need a help to clarify this one in particular. Suppose $M$ is a manifold with a coordinate chart ...
0
votes
2answers
44 views

Einstein Summation - does the following equality hold: $a_{ij} x_i y_j = a_{ij} y_i x_j$

Does equality hold when $x_i = y_i$ and $x_j=y_j,$ and $ i, j = 1, ..., n $.
0
votes
2answers
31 views

Einstein Summation: How do I show $a_{ij} (x_i + y_j) \not= a_{ij}x_i + a_{ij}y_j $?

Einstein Summation: How do I show $a_{ij} (x_i + y_j) \not= a_{ij}x_i + a_{ij}y_j $?
1
vote
0answers
40 views

What does this operator $\odot$ mean

I read this about the second fundamental form in Wikipedia and I’ve no idea what does $\odot$ mean? Does anybody know? $$II=-dN\cdot dP=\omega^3_1\odot\omega^1+\omega^3_2\odot\omega^2$$
0
votes
0answers
29 views

What is the notation for pull-back and push-forward of an exponential map?

So there is a nice notation for a one-parameter group of transformations $\Phi_t$ corresponding to its infinitesimal generator $\boldsymbol X$: $$\Phi_t = \exp \left(t \boldsymbol X \right)$$ But ...
5
votes
1answer
73 views

Space of Alternating $k$-Tensors Notation

I will be taking a Differential Geometry class in the Fall, so I decided to get somewhat of a head start by going through Spivak's "Calculus on Manifolds." Before reading, though, I saw the Addenda at ...
3
votes
1answer
97 views

Coincidence about nabla?

I was surprised to notice that gradient of function and Levi-Civita connection have the same notation, i.e. nabla sign $\nabla$. Moreover, extending any connection on tensors, one let it be ...
0
votes
0answers
56 views

Accepted symbol (or way of writing) “A is a subset of B or B is a subset of A”

I am looking for a concise way to write the statement "$A$ is a subset of $B$ or $B$ is a subset of $A$". The context is the Grassmannian and two elements $A,B\in G_k(\mathbb R^n)$ in it. The two ...
1
vote
1answer
47 views

Integral notation from cartesian from polar coordinates

Given an integral $$I=\int\limits_{\mathbb{R}^n} \cdot \; dx,$$ we can introduce polar coordinates, such that $$I=\int\limits_{\Bbb S^{n-1}} \cdot \; d\theta.$$ Another way to express the latter one ...
1
vote
1answer
41 views

Differential geometry notation (tensor vs. others)

As a physicist I am used to working with tensor notation in differential geometry, which is probably better for computation, but, e.g., I've seen $$R(u,v)w = \left( \nabla_u \nabla_v - ...
5
votes
1answer
82 views

Intuition for Integration of Differential Forms

In mathematics, we define $dx^i$ as linear functionals, when speaking of integration. However, in physics, we interpret $dx^i$ as very small quantities. There is nothing inherently small about a ...
2
votes
1answer
86 views

Lie derivatives and covariant derivatives (notation)

I am having troubling interpreting a particular expression in differential geometry. It arose in computing the Lie derivative along a unit normal, $n$, of the extrinsic curvature of a sub-manifold ...
2
votes
3answers
64 views

Why the name “umbilic”?

Umbilic points are points on a surface at which the principle curvatures of the surface are equal. "Umbilic(al)" refers to the navel/belly button. But why do we call these points so? What about the ...
6
votes
1answer
87 views

Tensor fields and vector bundles

Let $M$ be a differentiable manifold, $TM$ and $T^*M$ a tangent and cotangent bundle of $M$ and let $\Gamma (TM),\ \Gamma (T^*M)$ be spaces of smooth sections of $TM$ and $T^*M$. Let $T_s^r (M)$ ...
0
votes
1answer
30 views

choosing indices in tensor notation

I have the following operator, where $\rho$ is a scalar and $u$ is a vector: $$ \nabla (\rho u) - (\nabla \rho)u - u(\nabla \rho) $$ My book writes this in index notation as $$ \partial_\alpha(\rho ...
1
vote
1answer
58 views

Doubts with differential geometry notation in Frankel

This is from Frankel's The Geometry of Physics: Problem 2.3(2) Consider the tangent bundle to a manifold $M$. Show that under a change of coordinates in $M$, $\partial/\partial q$ depends ...
0
votes
1answer
30 views

Which set is this $I_p(p)\cdot \Gamma(E)$?

Let $\pi:E\rightarrow M$ be a smooth vector bundle and $p\in M$. Consider $$I_p(M)=\{f\in C^\infty(M): f(p)=0\},$$ and $\Gamma(E)$ the $C^\infty(M)$-module of smooth sections over $E$. Notice $I_p(M)$ ...
1
vote
1answer
113 views

Problem with notation: Laplacian on a manifold

In the Aubin's book "Nonlinear analysis on manifolds" the Laplacian operator on functions on some smooth manifold is defined by the formula $$ \Delta = -\nabla^\gamma\nabla_\gamma, $$ where ...
3
votes
3answers
323 views

Notation: subscript vs. superscript for coordinate vector fields

Some books write the coordinate vector fields with a subscript as $$\frac{\partial}{\partial x_i}$$ while some write it with a superscript as $$\frac{\partial}{\partial x^i}.$$ Is there a conceptual ...
0
votes
1answer
73 views

Standard notation for isometry group?

Let $M$ be a (semi) Riemannian manifold, is there a standard notation for the group of isometries on $M$? I would think $\mathrm{ISO}(M)$ would be appropriate, but I've never encountered a dedicated ...
1
vote
3answers
768 views

Why is the Jacobian matrix the transpose of what I would think it'd be/usefully be (total derivative is a synonym) (EDIT: I was a total wally)

I'm sorry this isn't a yes/no/am-I-right question but I seriously cannot see why the Jacobian/total derivative matrix is what it is? I am also using it as LaTeX practice (for maths) hence the barely ...
2
votes
0answers
94 views

“Product” bundle notation.

Let $\newcommand{\Spin}{\operatorname{Spin}}M$ and $M'$ be two manifolds, equipped with a principal $\Spin_n$ and $\Spin_{n'}$ bundle called $P$ and $P'$, respectively. Then there is an induced ...
1
vote
2answers
131 views

what is $C^{-\infty}(\mathbb{R})$

Thanks in advance. what is $C^{-\infty}(\mathbb{R})$? Is that the same as the "distribution" defined in differential geometry? It would be helpful if someone can describe it in another way ...
7
votes
2answers
548 views

Notation to work with vector-valued differential forms

What it the standard notation used while working with vector-valued differential forms? I tried using abstract index notation, for example denoting a $1$-form valued $2$-form as $P_{i[bc]}$, but I'm ...
1
vote
1answer
72 views

Notation and naming for two operations with $p$-form valued $n$-forms

While trying to answer my other question I found I never heard about vector-valued differential forms. I've been searching for them in various mathematical physics books, but didn't get too much. I'm ...
2
votes
2answers
100 views

Why parametrise a curve in this way (on the unit circle)?

I saw papers saying something like "let $\gamma:S^1 \times [0,T] \to \mathbb{R}^2$ parametrise a curve. The second interval above just makes it time dependent, but why parametrise (for fixed time) the ...
1
vote
1answer
104 views

differential and arc length notation question

Suppose $\alpha$ is a time dependent curve so that $\alpha:[0,T]\times I \to \mathbb{R} ^n$. I am a bit confused as to what the meaning of the expression $\partial_t(ds)$ is, where $ds = |\partial_x ...
6
votes
1answer
295 views

notation of derivation in differential geometry

I can't wrap my head around notation in differential geometry especially the abundant versions of derivation. Peter Petersen: Riemannian Geometry defines a lot of notation to be equal but I don't ...
1
vote
2answers
696 views

What does this symbol mean? (looks like a lower-right corner — subject: manifolds theory)

Below I have posted an excerpt of Lee's Book "Introduction to Smooth Manifolds" (page 371). I don't know what the symbol means that looks like a lower-right corner, and I cannot find it via the index, ...
5
votes
4answers
556 views

What does the symbol $\operatorname{Tr}$ in the Yang-Mills action mean?

I find that many authors write the Yang-Mills action as follows: $$\mathcal{J}= \int \operatorname{Tr}(F \wedge \star F).$$ I have yet to find a formal description of the symbol $\operatorname{Tr}$ ...
3
votes
1answer
348 views

Christoffel Symbol - what does a comma mean in the footer?

I am trying to understand the expression for Scalar curvature in terms of the Christoffel symbols. This is given on Wikipedia by \begin{equation} S = g^{ab}(\Gamma^c_{ab,c} - \Gamma^c_{ac,b} + ...
5
votes
1answer
350 views

idea of the star position in pullback, pushforward notation

i would like to know if there is some idea behind the position of the star in the pullback, pushforward notation or if it is just some notation without background? What's the reason for the star to ...
1
vote
1answer
371 views

The Dual Pairing

My understanding from the reading the Wikipedia article on Dual Pairs is that a dual pair is comprised of two vector spaces $X$ and $Y$ over a field $\mathbb{K}$ together with a nondegenerate ...
3
votes
0answers
147 views

bar index notation

For complex manifolds , people usually write the first fundamental form as $ds^2=g_{a\bar{b}}dz^ad\bar{z}^b$ (at least physicists) with a bar over the second index of the metric, but don't usually ...